Not sure whether this is what you have in mind, but there is Abstract Model Theory, where one certainly considers classes that are not axiomatizable in first order logic (or in several versions of infinitary logic). The standard reference is "Model-theoretic logics", projecteuclid.org/…
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Andres CaicedoJun 20 '12 at 20:19

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But you seem to have a more general notion of model (structure) in mind. You may then want to look into Algebraic Logic, en.wikipedia.org/wiki/Algebraic_logic and particularly into the topic of algebraic semantics.
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Andres CaicedoJun 20 '12 at 20:22

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This depends strongly on the extent of your "etc." For example, classical FOL and classical propositional logic are special cases of multi-sorted classical FOL; propositional logic is the case of 0 sorts. A better answer is a pointer to the book "Model-Theoretic Logics" (Springer, 1985) edited by Jon Barwise and Sol Feferman, particularly the first two chapters, "Model-Theoretic Logic: Background and Aims" by Barwise, and "Extended Logics: The General Framework" by Heinz-Dieter Ebbinghaus. The general idea is that an abstract logic determines, for each vocabulary (= language = signature) a collection of sentences and a satisfaction relation between structures and sentences, all subject to some very basic axioms (e.g., isomorphic structures satisfy the same sentences).

What you are after is institution-independent model theory, for which Diaconsecu has a recent textbook account. Here the bare concept of an institution is basically that of a logical system absent particular distinguishing features; things such as classical first-order logic, higher-order logic and intuitionistic logic are then instances of institutions.

In general you can pick any set of properties regarding the model theory of logics you are interested in and create an abstract theory about them. Most of these will not be interesting for various reasons but some will be. An example of nice class of model theoretic properties is those chosen for Abstract Elementary Classes. You may want to check John Baldwin's slides to get a feeling of the model theory that can be derived using such simple model theoretic axioms.